Quarterly payment of 5,000 for 10 years with interest rate of 2% compounded annually

To determine the present value of a series of quarterly payments of $5,000 made over 10 years with a 2% interest rate compounded annually, we’ll use the formula for the present value of an annuity.

Since the payments are quarterly, let’s break down the solution carefully:

1. **Convert the annual interest rate to an effective quarterly rate**:
   - Annual interest rate = 2% (or 0.02 in decimal form)
   - Effective quarterly interest rate $ i_{\text{quarter}} $:
     \[
     i_{\text{quarter}} = \left(1 + \frac{0.02}{1}\right)^{\frac{1}{4}} - 1 \approx 0.004962 \quad \text{(or about 0.4962% per quarter)}
     \]

2. **Calculate the total number of payments**:
   - Since payments are quarterly for 10 years, the total number of payments $ n $ is:
     \[
     n = 10 \times 4 = 40
     \]

3. **Use the Present Value of an Annuity Formula**:
   The formula for the present value $ PV $ of an annuity is:
   \[
   PV = P \times \frac{1 - (1 + i)^{-n}}{i}
   \]
   where:
   - $ P = 5,000 $ (payment per period)
   - $ i = 0.004962 $ (quarterly interest rate)
   - $ n = 40 $ (number of payments)

4. **Substitute and Solve**:
   \[
   PV = 5,000 \times \frac{1 - (1 + 0.004962)^{-40}}{0.004962}
   \]
   Calculating this yields the present value.

Let me calculate the exact amount.The present value of the quarterly payments of $5,000 over 10 years at a 2% annual interest rate compounded annually is approximately **$180,993.53**.

Would you like a detailed breakdown or have any questions?

Here are some related questions that might help further:

1. How would the present value change if payments were made monthly instead of quarterly?
2. What if the interest rate were compounded quarterly instead of annually?
3. How would the future value of this annuity differ with the same parameters?
4. What would be the effect of a higher interest rate on the present value?
5. How can we calculate the equivalent annual payment that yields the same present value?

**Tip:** When working with annuities, adjusting the compounding frequency affects the effective interest rate, which in turn changes the present or future values.

Discover how to calculate the present value of quarterly payments of $5,000 for 10 years with a 2% interest rate compounded annually. This detailed solution involves key financial concepts and formulas, yielding an approximate present value of $180,993.53.

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